One-fold rotation axis

In the Cih point group, there is a one-fold rotation axis plus horizontal symmetry (horizontal mirror). Note also the difference between Cih and... 

The symmetry plane and the rotation axis are symmetry elements. If a figure has a symmetry element, it is symmetrical. If it has no symmetry element, it is asymmetrical. Even an asymmetrical figure has a one-fold rotation axis or, actually, an infinite number of onefold rotation axes. 

Ci There are no symmetry elements except the one-fold rotation axis, or identity, of course. Ci symmetry is asymmetry. Examples are ... 

Dnh The series can be continued by analogy. There will be one //-fold rotation axis, one symmetry plane perpendicular to it, and n symmetry planes which include the //-fold axis. When n is even, there are two sets of symmetry planes. One set is rotated relative to the other set by (180///)°. The angle between the planes within each set is (360/n)°. When n is odd, the angle between the symmetry planes is (180///)°. 

The one-fold rotation axis, shown in Figure 1.11 on the left, rotates an object by 360", or in other words converts any object into itself, which is the same as if no symmetrical transformation had been performed. It is the only symmetry element, which does not generate additional objects except the original. One-fold rotation axis, also known as the "identity" or unity operation, is used in crystallography for logical completeness in the treatment of symmetry as will be shown later (see sections 1.6 and 1.7). Furthermore, a one-fold rotation axis is always present in any object or in any crystal structure. 

Figure 1.11. One-fold rotation axis (left) and center of inversion (right)...

This example not only explains how the two symmetry elements interact, but it also serves as an illustration to a broader conclusion deduced above any two symmetry operations applied in sequence to the same object create a third symmetry operation, which applies to all symmetrically equivalent objects. Note, that if the second operation is the inverse of the first, then the resulting third operation is unity (the one-fold rotation axis, 1). For example, when a mirror plane, a center of inversion, or a two-fold rotation axis are applied twice, all result in a one-fold rotation axis. 

As established before, the associative law holds for symmetry groups. Returning to the example in Figure 1.16, which includes the mirror plane, the two-fold rotation axis, the center of inversion and one-fold rotation axis (the latter symmetry element is not shown in the figure and... 

Dy D, D-j,. . ., D . This series can be continued by analogy. It is characterized by one -fold rotation axis and n twofold rotation axes perpendicular to the -fold axis. 

Fig. 3.11 The three minimum configuration for the regular hexagon of points (a) has a one-fold rotational axis of symmetry (b) has a three-fold rotational axis of symmetry (c) has a two-fold rotational axis of symmetry, (a) has the smallest path length.

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